Evaluate 1/(cos(pi/9)^2)-1/(cot(pi/9)^2)

$\frac{1}{\cos^{2} (\frac{π}{9})} - \frac{1}{\cot^{2} (\frac{π}{9})}$

Step 1

Simplify terms.

Simplify each term.

Rewrite $1$ as $1^{2}$ .

$\frac{1}{\cos^{2} (\frac{π}{9})} - \frac{1^{2}}{\cot^{2} (\frac{π}{9})}$

Rewrite $\frac{1^{2}}{\cot^{2} (\frac{π}{9})}$ as ${(\frac{1}{\cot (\frac{π}{9})})}^{2}$ .

$\frac{1}{\cos^{2} (\frac{π}{9})} - {(\frac{1}{\cot (\frac{π}{9})})}^{2}$

Rewrite $\cot (\frac{π}{9})$ in terms of sines and cosines.

$\frac{1}{\cos^{2} (\frac{π}{9})} - {(\frac{1}{\frac{\cos (\frac{π}{9})}{\sin (\frac{π}{9})}})}^{2}$

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (\frac{π}{9})}{\sin (\frac{π}{9})}$ .

$\frac{1}{\cos^{2} (\frac{π}{9})} - {(1 \frac{\sin (\frac{π}{9})}{\cos (\frac{π}{9})})}^{2}$

Write $1$ as a fraction with denominator $1$ .

$\frac{1}{\cos^{2} (\frac{π}{9})} - {(\frac{1}{1} \cdot \frac{\sin (\frac{π}{9})}{\cos (\frac{π}{9})})}^{2}$

Simplify.

Rewrite the expression.

$\frac{1}{\cos^{2} (\frac{π}{9})} - {(1 \frac{\sin (\frac{π}{9})}{\cos (\frac{π}{9})})}^{2}$

Multiply $\frac{\sin (\frac{π}{9})}{\cos (\frac{π}{9})}$ by $1$ .

$\frac{1}{\cos^{2} (\frac{π}{9})} - {(\frac{\sin (\frac{π}{9})}{\cos (\frac{π}{9})})}^{2}$

Apply the product rule to $\frac{\sin (\frac{π}{9})}{\cos (\frac{π}{9})}$ .

$\frac{1}{\cos^{2} (\frac{π}{9})} - \frac{\sin^{2} (\frac{π}{9})}{\cos^{2} (\frac{π}{9})}$

Combine the numerators over the common denominator.

$\frac{1 - \sin^{2} (\frac{π}{9})}{\cos^{2} (\frac{π}{9})}$

Step 2

Apply pythagorean identity.

$\frac{\cos^{2} (\frac{π}{9})}{\cos^{2} (\frac{π}{9})}$

Step 3

Cancel the common factor of $\cos^{2} (\frac{π}{9})$ .

Cancel the common factor.

$\frac{\cos^{2} (\frac{π}{9})}{\cos^{2} (\frac{π}{9})}$

Rewrite the expression.

$1$